In the case of a prime \(p\) ,
we will now first consider the more general situation of class functions \(\chi ^{(p)}\)  and relate these to \(p\) -modular decomposition numbers
(or \(p\) -decomposition numbers, for short).
For more details on the background
of the modular theory, we refer the reader to
the textbooks [1]}, [2]}, [3]}, [4]}.
For the convenience of readers less familiar with the modular theory,
and also to fix some notation on the way,
we give a brief introduction to the theory.
